How to put 2 parts of 1 object on different layers?
You cannot move different parts of the same object to separate layers.

As mentioned by sambler, what you are trying to do is possible using separate materials to generate masks.

Quoting from my answer here:


  This can be done . . . using an ID Mask node:
  
  . . .
  1. Enable Material index in Render layers > Passes, and set the pass index for the material:
  
  
  Use Materials > Settings > Pass > index to set the pass index of the  material in Cycles:
  
  
  Or in Blender internal use Materials > Options > Pass index:
  
  
  
  
  
  
  
  Add an ID Mask node and set the Index to 1.
  Plug the IndexOB output from the Renderlayer node into the ID Mask node. This will provide a mask for the [material]:
  
  


This will give you a black and white mask of that material, which you can use to separate, combine, etc. in the compositor using mix nodes.